Answer to Question #270567 in Discrete Mathematics for migs

Question

A weighted voting system for voters A, B, C, D, and E is given by {35: 29, 11, 8, 4, 2}. The weight of voter A is 29, the weight of voter B is 11, the weight of voter C is 8, the weight of voter D is 4, and the weight of voter E is 2.

a. What is the quota?

b. What is the weight of the coalition {A, D, E}?

c. Is {A, D, E} a winning coalition?

d. Which voters are critical voters in the coalition {A, C, D, E}?

e. How many coalitions can be formed?

f. How many coalitions consist of exactly two voters?

Answer 1

Given weighted voting system is: { 35: 29,11, 8, 4, 2 }

(a) Quota of the given voting system is 35.

(b) Weight of the coalition {A, D, E} = 29+4+2= 35

(c) Yes, {A, D, E} is a winning coalition because its weight is equal to quota.

(d) Critical voter in the coalition {A, C, D, E) is voters A.

(e) Total number of coalitions formed is 2ⁿ - 1 where n is number of voters. Therefore, 

Total number of coalitions formed = 2⁵ — 1 = 32 - 1 = 31 

(f) There are 10 coalitions consist of exactly two voters. i.e. {A,B}, {A,C}, {A,D}, {A,E}, {B,C}, {B, D}, {B, E}, {C, D}, {C, E} and {D, E)